Attitude toward imprecise information

This paper presents an axiomatic model of decision making under uncertainty which incorporates objective but imprecise information. Information is assumed to take the form of a probability-possibility set, that is, a set P of probability measures on the state space. The decision maker is told that the true probability law lies in P and is assumed to rank pairs of the form (P, f) where f is an act mapping states into outcomes. The representation result delivers maxmin expected utility at each probability-possibility set. Furthermore, the set over which the minimum is taken is linked to the information: there is a mapping that gives for each probability-possibility set the set of selected probability distributions. This allows both expected utility when the set is reduced to a singleton and extreme pessimism when the decision maker takes the worst case scenario in the entire probability-possibility set. We define a notion of comparative imprecision aversion and show it is characterized by inclusion of the sets of revealed probability distributions, irrespective of the utility functions that capture risk attitude. We also identify an explicit attitude toward imprecision that underlies usual hedging axioms. Finally, we characterize, under extra axioms, a more precise functional form, in which the set of selected probability distributions is obtained by (i) solving for the “mean value” of the probability-possibility set, and (ii) shrinking the probability-possibility set toward the mean value to a degree determined by preferences.